## Abstract

The charge-membrane voltage (*Q-V*) distribution of wild-type rabbit Na^{+}/glucose transporter (rSGLT1) expressed in *Xenopus* oocytes was investigated in the absence of glucose, using the two-electrode voltage-clamp technique. Although this distribution is generally believed to be well represented by a two-state Boltzmann equation, we recently provided evidence for the existence of at least four states (Krofchick D and Silverman M. *Biophys J* 84: 3690–3702, 2003), confirming an earlier finding for human SGLT1 (Chen XZ, Coady MJ, and Lapointe JY. *Biophys J* 71: 2544–2552, 1996). We now extend our study of rSGLT1 pre-steady-state currents, employing high-resolution measurement and analysis of the *Q-V* distribution. A ramp, instead of a step, voltage change was used to prevent saturation of the apparatus in the first ∼1 ms. Transient currents were integrated out to 150 ms, instead of the standard 50–100 ms. Measurements were taken every 10 mV instead of the standard 20 mV. The *Q-V* distribution was fit with a two-, three-, and four-state Boltzmann equation and was described best by the three-state equation. The three-state fit produced two valences of 0.45 and 1.1 at two *V*_{0.5} values of −48 and −7.7, respectively. Our findings are critically compared with other published studies and the differences are discussed. An implication of the three-state fit is that the turnover rate of rSGLT1 is 34 s^{−1}, i.e., 54% greater than previously reported (22 s^{−1}). Our new findings support the concept that the sugar-free model of SGLT1 is more complex than generally accepted, most likely involving a minimum of four transition states.

- SGLT1
- Boltzmann distribution
*Xenopus*oocyte- sodium/glucose cotransport
- two-electrode voltage clamp

ion-coupled solute cotransporters are expressed in all tissues and subserve diverse cell physiological functions. These surface constituents are essential determinants of the cell's capacity to regulate its internal environment by maintaining complex secondary active transport functional networks. In mammals, this involves coupling of Na^{+}-dependent solute symport and antiport proteins with the transmembrane Na^{+} electrochemical gradients created by metabolically driven Na^{+}-K^{+}-ATPase (i.e., Na^{+} pump). Epithelial secretion and absorption are examples of the emergent properties that result from this functional organization.

Despite considerable variation in individual structure and function, ion-coupled solute cotransporters exhibit common behavioral characteristics. This has led to the concept that this class of membrane proteins has evolved specific domains that enable the functional properties central to their physiological roles in mediating secondary active transport, namely, stoichiometric coupling of cotransported species, and voltage sensing.

Probably the best studied example of ion-coupled cotransporters is the family of proteins responsible for cotransport of Na^{+} and d-glucose, especially the isoform for intestinal absorption of glucose and galactose, SGLT1. This carrier molecule harnesses the electrochemical gradient of Na^{+} to move extracellular Na^{+} and glucose inside cells at a ratio of two Na^{+} ions per glucose molecule. Despite intensive investigation of SGLT1 since it was successfully cloned in 1987 (5), the fundamental underlying molecular/atomic mechanism(s) responsible for its biological activity remain unknown. However, progress has been made in characterizing a plethora of kinetic constants by using the *Xenopus* oocyte expression system in conjunction with voltage-clamp techniques, for example; affinity constants have been found for various ligands (Na^{+}, glucose, phloridzin, etc.); from the Hill coefficient we know the stoichiometry between Na^{+} and sugar; steady-state current-voltage (*I*-*V*) curves give insight into sugar uptake and the turnover rate; fitting transient current with exponential decays characterizes transition states and enumerates them; and charge-membrane voltage (*Q-V*) curves tell us about the voltage sensitivity of the system and state occupancies.

Over the years our understanding of the functional properties of the transporter has grown; however, we have reached a point where we are realizing that the current models of SGLT1 (common 6- and 8-state systems) are too simple to account for certain phenomena and that more complicated models are needed. Examples include the observation of two conformational transitions of the unloaded carrier compared with the previous assumption of a single conformational transition (3, 7), the observation of a conformational change following sugar binding but independent of sugar transport (13, 14), and the presence of a conformational change after Na^{+} binding but before sugar binding (11). Exploring these phenomena requires the advent of more detailed kinetic models and the incorporation of current kinetic data into these models. Recently, using the two-electrode voltage-clamp setup, we demonstrated that the transient currents of rabbit SGLT1 (rSGLT; observed in the absence of glucose) are composed of a minimum of three exponential decays (two more than previously assumed) and that, in turn, the system under investigation contains at least four states (see Fig. 1). These four states, C_{i}, are believed to be the following: C_{1}, cotransporter with Na^{+} bound at the extracellular surface; C_{2}, cotransporter facing the extracellular environment without Na^{+} bound; C_{3}, an intermediate conformation of the unloaded carrier; and C_{4}, cotransporter facing the intracellular environment (3, 7). A common and useful experimental measure for characterizing the voltage-sensing properties of SGLT1 in all its forms (rabbit, human, rat, and various site-directed mutants, etc.) is a *Q*-*V* curve that is well modeled by a Boltzmann distribution. Qualitatively, the *Q*-*V* curve depicts the distribution of the transporters among the system's states (through *Q*) as a function of the energy of the system (membrane potential, *V*). Quantitatively, the *Q*-*V* distribution (when fit with an appropriate Boltzmann equation) provides a direct measure of the number of states in the system and the number of expressed transporters on the oocyte, *N*, in addition to the system's voltage sensitivity, *z*. Also, both *N* and *z* are used to calculate turnover rate, *k*.

All previous investigations of SGLT1 have modeled the *Q-V* curve with a Boltzmann distribution representing a two-state system (rabbit, human, and various site-directed mutants of both) because this model provides reasonably good visual fits to the data despite common knowledge at the time clearly recognizing a three- and, possibly, four-state system (3, 4, 6, 8–11, 13, 15–17, 20). In an attempt to bring our interpretation of the experimental data in line with our current understanding of the SGLT1 model, we present in this article the results of an in-depth measurement and analysis of the rSGLT1 *Q-V* distribution at a concentration of 100 mM external Na^{+}. A number of changes have been made to the standard protocol for measuring the *Q-V* distribution: *1*) the standard step change in membrane potential has been replaced by a 5-ms ramp to prevent saturation of the measuring apparatus in the first ∼1 ms, caused by large membrane capacitive currents, which was resulting in an underestimation of *Q*; *2*) data points were collected every 10 mV, instead of every 20 mV, to bring out finer structures in the distribution; *3*) we found that the transient currents persisted for ∼125 ms and consequently changed our integration domain from 100 to 150 ms; and *4*) the *Q*-*V* data were fit with a two-, a three-, and a four-state Boltzmann equation. Residual and χ^{2}/df (df, degrees of freedom) data clearly showed that the three-state Boltzmann equation was superior to the two-state equation. There were, however, too many degrees of freedom in the four-state equation to fit the data, leading us to conclude that there is insufficient detail in the 100 mM Na^{+} Boltzmann curve to fit the four-state equation and that the three-state Boltzmann equation is the best approximation (other Na^{+} concentrations may give better results). Our experimental results and analysis of the rSGLT1 *Q*-*V* distribution differ significantly from those previously published. New insights presented here expand our understanding of the transition states that describe the pre-steady-state currents of rSGLT1.

## MATERIALS AND METHODS

#### Molecular biology.

The multicloning site of the eukaryotic expression vector pMT3 (provided by the Genetics Institute, Boston, MA) was removed by digestion with *Pst*I and *Kpn*I, and the cDNA of rSGLT1 (kindly provided by M. Hediger, Harvard Institutes of Medicine, Boston, MA) was subcloned into the remaining *Eco*RI site (8).

#### Oocyte preparation and injection.

Oocyte extraction was performed in conformity with protocols approved by the University of Toronto Animal Care Committee. *Xenopus laevis* frogs were anesthetized with a 0.17% aqueous solution of 3-aminobenzoic acid ethyl ester. Extracted oocytes were digested for 25–90 min with 2 mg/ml type IV collagenase (Sigma, Oakville, ON, Canada) prepared in modified Barth's saline (MBS) supplemented with MgCl_{2} (MBS-Mg^{2+} solution: 0.88 mM NaCl, 1 mM KCl, 2.4 mM NaHCO_{3}, 1 mM MgCl_{2}, and 15 mM HEPES-Tris, pH 7.4). For all storage after digestion, oocytes were kept at 16–18°C in Leibovitz's L-15 solution (Sigma) supplemented with 0.08 mg/ml gentamicin, 0.736 g/l l-glutamine, and 10 mM HEPES-NaOH, pH 7.4. The day after digestion, 9.2 nl of 60 ng/μl cDNA of rSGLT1 was injected into the nucleus, via the animal pole, of defolliculated oocytes.

#### Two-microelectrode voltage clamp.

Voltage-clamp experiments were performed 4–6 days after injection with the use of a GeneClamp 500 amplifier and pCLAMP 6.0 data acquisition software (Axon Instruments, Union City, CA). Electrode tips, fabricated from 150-μm borosilicate glass capillary tubes, were pulled using the model p-97 Flaming/Brown micropipette puller. Tips were filled with 3 M KCl. Between measurements, oocytes were constantly perfused at ∼3.5 ml/min with voltage-clamping solution (100 mM NaCl, 2 mM KCl, 1 mM MgCl_{2}, 1 mM CaCl_{2}, and 10 mM HEPES-Tris base, pH 7.4). Only oocytes with resting potentials more hyperpolarizing than −30 mV were used.

In an oocyte expressing SGLT1, measured currents consist of nonspecific components (generated by capacitive currents and endogenous transporters and channels) and an SGLT1-specific component that is inhibitable by phloridzin. For all experiments, SGLT1-specific currents were isolated by subtracting currents measured immediately before and after the addition of saturating 0.2 mM phloridzin, with the assumption that both specific and nonspecific currents are measured in the absence of phloridzin, whereas only nonspecific currents are detected in the presence of phloridzin. Because the *K*_{i} for phloridzin is 1–10 μM (unpublished data), 200 μM phloridzin completely blocks SGLT1-specific currents. This has been confirmed by plots of the integral of transient currents in the presence of 200 μM phloridzin, which are linear between −150 and 70 mV at both 10 and 100 mM Na^{+} (data not shown), suggesting that the transient current response is solely capacitive over this voltage range (i.e., transporter currents are completely inhibited by phloridzin).

#### Protocol for measuring the Q-V distribution.

A typical protocol for measuring the *Q-V* distribution is shown in Fig. 2. The family of transient currents, produced by stepping the membrane potential from a fixed prestep potential (usually −50 mV) to a range of poststep potentials (−150 to 70 mV in 20-mV increments), is integrated to give net charge movement, *Q*, as a function of the poststep potential. A close-up of the beginning of the transients, measured in the absence and presence of phloridzin, clearly shows saturation of the measuring apparatus in the first millisecond, caused by large membrane capacitive currents (Fig. 3, *A* and *B*). After the phloridzin subtraction protocol (Fig. 3*C*), sections of the transients are missing where saturation occurred.

To address the problem of saturation, we used a 5-ms ramp protocol (Fig. 4). This protocol reduced the transient currents to within an acceptable range for the apparatus. A 5-ms ramp was chosen because shorter ramp times tended to saturate the apparatus, depending on oocyte expression levels, whereas longer ramp times decreased the magnitude of the currents, reducing the signal-to-noise ratio. Transients were measured out to 150 ms, at which time they were found to be at steady state (see *Duration of transient currents* in results). Before integration, the baseline was adjusted by the average current between 140 and 149 ms. The data were not filtered. The number of traces was increased from 12 to 23 with recordings made in 10-mV increments (instead of 20 mV) from −150 to 70 mV. The sampling interval was 25 μs. It is important to point out that the ramp protocol is equivalent to the step protocol for measuring the *Q-V* distribution (assuming there is no saturation of the measurement apparatus) because the system is conservative (i.e., the work done by a force on the system is path independent). In other words, the shape of the voltage clamp is inconsequential, such that net charge transfer, *Q*, is only a function of the initial and final voltages (see *Eqs. 11* and *15* in appendix).

There is a possible alternate approach for preventing saturation that we mention here for completeness. In many patch-clamp amplifiers there is a feature that allows for the modeling of the cell membrane by an RC (resistive/capacitive) circuit. Transient current from the model circuit is subtracted dynamically from the membrane current and thus prevents saturation of the apparatus (1). Unfortunately, this feature is not supported by most two-electrode voltage-clamp amplifiers and, to the best of our knowledge, is offered by only a single manufacturer.

#### Model assumptions and limitations.

There are a number of assumptions and limitations in the Boltzmann fitting process that may not be immediately obvious: *1*) there is a Na^{+} leak pathway between C_{1} and C_{4} in Fig. 1 that we have ignored [as is common practice for SGLT1 transient studies (3, 4, 7, 10, 13)] because it represents only a small fraction of the system currents and because it leads to a five-state system that is beyond our ability to detect, given that a three-state system was the maximum we could model; *2*) there may be some processes that occur in parallel to the model in Fig. 1, in which case an appropriate model would sum the Boltzmann equations of all parallel systems; *3*) if there are multiple systems in parallel with similar rate constants, then it would be impossible to distinguish them from fitting the *Q-V* distribution, because their distributions would overlap (e.g., the sum of 2 identical 2-state Boltzmann distributions with *z* = 1 is fit exactly by a single 2-state Boltzmann distribution with the same parameters so that the system would appear to have a valence of 1 when in fact it is 2); *4*) high-order systems may often have a *Q-V* distribution that is virtually identical to the distribution produced by a much lower order system, therefore making them indistinguishable {e.g., the 2-state Boltzmann equation *Q*(*V*) = 0.5 − 1/(1 + *e*^{0.63V}) and the 3-state equation *Q*(*V*) = [0.5 − (1 + 0.5*e*^{0.5V})]/(1 + *e*^{0.5V}+ *e*^{0.5V}*e*^{0.5V}) are practically identical}.

#### Normalizing the Q-V distribution.

Before *Q-V* data between two or more oocytes can be compared, the data must be normalized to account for variation in expression levels among the eggs. However, the normalization constant, *N*, which represents the number of expressed transporters on an oocyte, is itself obtained by fitting the *Q-V* data with a Boltzmann equation (see *Eqs. 11* and *15*) and is therefore dependent on the order of the Boltzmann equation used and the accuracy of the data. Therefore, to be able to determine the optimum Boltzmann equation, transient data from multiple oocytes (*n* = 18) were amalgamated into a single data set via point-by-point addition of the currents. This amalgamated data set had a large signal-to-noise ratio. Because the signal-to-noise ratio increases with the square root of the number of data sets combined, if we ignore variations in expression level between oocytes (as a first approximation), then the signal-to-noise ratio after amalgamation is ∼4.2 times larger [SNR_{2} = (18)^{1/2}·SNR_{1} ≈ 4.2·SNR_{1}, where SNR is signal-to-noise ratio]. The combined *Q-V* data were fit with a two-, a three-, and a four-state Boltzmann equation (see *Eqs. 11* and *15* for 2- and 3-state equations, respectively).

## RESULTS

#### Duration of transient currents.

To determine the duration of the transient currents, we collected a single aggregate data set of transient currents from multiple oocytes (*n* = 18; see *Normalizing the Q-V distribution* in materials and methods) and compared *Q-V* distributions in 25-ms increments (Fig. 5*A*) along with the diminishing returns in *Q* collected from each 25-ms increment (Fig. 5*B*). Figure 5*A* shows that the vast majority of charge was collected in the first 75 ms; however, Fig. 5*B* shows that the transients continued out to ∼125 ms before they vanished. This is a reasonable duration for the transients given that the slowest time constant reported for rSGLT1 is ∼45 ms (7) and that ∼95% of the current, from the slowest transient, was collected over a period of three decays (45 × 3 = 135 ms). Therefore, 150 ms is sufficient to collect all of the observable charge movement in the system.

Note that 150 ms is a much longer time domain than what has been used previously for rSGLT1. In 1992, Parent et al. (18) measured the transients for 70 ms, whereas in 1994, Panayotova-Heiermann et al. (16) used an interval of 100 ms and then in 1995 claimed that the transients lasted for only 50 ms (17). This variability in time domains is likely indicative of a poor ability to gauge the transient durations by “eye.” Thus it is important to employ a more empirical measure, like that shown in Fig. 5, because all of the Boltzmann parameters (*z*, *V*_{0.5}, *Q*_{max}, and *N*) are used to compare and contrast different species and mutants of SGLT1, and incomplete windows of observation can have a significant impact on all of these parameters.

#### Two- and three-state Boltzmann fits.

An aggregate *Q-V* distribution (*n* = 18) for wild-type rabbit is shown in Fig. 6*A* along with fits to a two- and a three-state Boltzmann equation (see *Eqs. 11* and *15*). Fit parameters are summarized in Table 1. The two-state Boltzmann fit has a χ^{2}/df of 22.7 × 10^{5} compared with that of 2.08 × 10^{5} for the three-state fit, a reduction by a factor of 11. Differences between the two- and three-state fits are seen more clearly on a plot of the residuals (Fig. 6*B*), which clearly shows the improvement made by using the three-state equation. For a good fit, the residuals should mimic random noise. The ability to predict consistently, and with confidence, where the two-state Boltzmann fit will under- and overfit the data indicates that the equation is inappropriate. The three-state fit residuals, however, appear to contain only noise. We were unable to include a four-state Boltzmann fit because there were too many degrees of freedom in the four-state equation for the given data. This is evident from the three-state residuals (Fig. 6*B*), which leave little to no room for improvement by a higher order equation.

The impact of the third-order Boltzmann fit can be summarized as the introduction of a second *V*_{0.5} (*V*_{10.5}) at hyperpolarizing potentials (−48 mV) with a valence (*z*_{1}) of 0.45. There was almost no change in *V*_{20.5}} and only a slight increase in *z*_{2} from 0.97 to 1.1. Note that the apparent valence, *z*_{app} = Σ*z*_{i}, is much larger for the three-state fit (1.45 vs. 0.97). This is an important finding, because *z*_{app} represents the net movable charge in the system and is a fundamental parameter in modeling. Although there is no change in *Q*_{max}, the number of predicted transporters, *N*, decreases by 35% from 9.4 × 10^{11} to 6.1 × 10^{11} carriers for the two- and three-state systems, respectively.

## DISCUSSION

#### Model discrimination.

When we recently analyzed the exponential decays that comprise the transient currents of rSGLT1 (7), we discovered that little attention was paid in the literature to model discrimination during the curve-fitting process (3, 4, 9–11, 13, 15–19). By this, we mean that despite there being a number of plausible models, there was an exclusive reliance on visual correlation between data and fits with no discussion of more empirical comparative data such as residuals, χ^{2}/df, or other goodness-of-fit statistics. This is even more significant given that the order of the fitting equations was lower than the known complexity of the system. As our understanding of the SGLT1 family of transporters grows and improvements are made in experimental techniques, it is increasingly important to quantify our confidence in the equations chosen to model the data. With regard to the *Q-V* distribution of SGLT1, we found that there, too, was an exclusive reliance on visual similarities between two-state Boltzmann fits and the data (3, 4, 6, 8–11, 13, 15–17, 19, 20). This can be attributed, in part, to a small signal-to-noise ratio and relatively good visual fits. Although it was possible to perform two- and three-state Boltzmann fits successfully for single oocyte data, we found that the signal-to-noise ratio could be significantly increased by summing multiple data sets together. Unfortunately, we were unable to present a mean data set with standard deviations because of the nature of the normalization procedure for the data (see *Normalizing the Q-V distribution* in materials and methods).

Of the three models/equations tested (2-, 3-, and 4-state systems), we found that both the residuals and the χ^{2}/df were good indicators for distinguishing between the models. The two-state Boltzmann fit residuals contained distinct nonrandom components, whereas the three-state residuals did not (Fig. 6*B*). The χ^{2}/df decreased by an order of magnitude (from 22.7 × 10^{5} to 2.08 × 10^{5}) when the three-state equation was used instead of the two-state equation (Table 1). We were, however, unable to find a suitable four-state Boltzmann fit to the data, even though we know that the system contains at least four states (3, 7). In fact, a four-state fit is most likely impossible, given that the three-state residuals seem to contain only random noise. We attribute this to an insufficient quantity of defining features in the data to be able to differentiate between a three- and a four-state model. In other words, it is possible that some four-state Boltzmann distributions have the same form as some three-state distributions and, therefore, from a fitting perspective they cannot be distinguished. It might be possible to find an appropriate set of four-state Boltzmann parameters if Na^{+}-dependent data were considered. Although the three-state Boltzmann equation may not reflect the true system, it is a best approximation and a considerable improvement over a two-state fit. For example, assuming we could obtain four-state Boltzmann parameters for rSGLT1, they would contain one additional *z* and *V*_{0.5} value so that the four-state *z*_{app} might be larger than the three-state *z*_{app}, but at the very least, the three-state *z*_{app} is a much closer estimate than the two-state *z*_{app}. This reasoning is supported by the work of Zampighi et al. (20), who estimated *z*_{app} to be ∼3.5 by first measuring *Q*_{max} and then estimating *N* by measuring particle density using the freeze-fracture method. This, of course, suggests that the upper limit of *z*_{app} is higher than that found in the present study and supports our position that although a three-state Boltzmann equation is able to account for the distribution of the *Q-V* curve, a higher order system is more likely.

#### Comparison with other studies of rSGLT1 Q-V distribution.

There is one major publication by Hazama et al. (4) of wild-type rSGLT1 kinetics that, among other things, investigates the *Q-V* distribution. In that study, the *V*_{0.5} value varied widely between −5 and 30 mV depending on the oocyte (4), contradicting an earlier finding in which the *V*_{0.5} value was specified as 0.5 ± 0.1 (16). In the present study, we also found no substantial variation in the *V*_{0.5} value. Using a two-state Boltzmann equation, we found that *V*_{0.5} = −5.6 ± 5.5 (*n* = 18). Moreover, a survey of the literature reveals that for rat, human, and a number of human and rabbit mutants, the *V*_{0.5} standard deviation varies between 0.5 and 5 mV (8, 9, 11, 13, 15, 17). Therefore, although we cannot explain the fluctuation reported by Hazama et al., we can safely say that this observation is specific to this study and that the *V*_{0.5} value of rSGLT1 is relatively stable.

The two-state Boltzmann *V*_{0.5} values reported by Hazama et al. (4) are also considerably more depolarizing than those found in the present, as well as other, studies (6, 16), i.e., Hazama et al. reported that *V*_{0.5} = 10 mV at 25°C and 34 mV at 20°C. Interpolating between these temperatures to 22°C, the temperature used for experiments in the present study, Hazama et al. would have predicted a Boltzmann equation with a *V*_{0.5} value somewhere between 10 and 34 mV, instead of the *V*_{0.5} value of −6 mV actually determined. Thus the two-state Boltzmann *V*_{0.5} value presented by Hazama et al. is 16–40 mV more positive that that found by us (*V*_{0.5} = −6) and 10–34 mV more positive than that reported by others (6, 16) (*V*_{0.5} = 0.5 and −3, respectively). The only way we have been able to explain this discrepancy is by noting that the *V*_{0.5} value shifts to the right as the time of integration is decreased (see Fig. 5). From this observation we have been able to correlate the *V*_{0.5} values as reported by Hazama et al. with an integration time of no longer than 70 ms (no interval is specified in Ref. 4). This deduced time interval makes sense given that the slowest time constant published by Hazama et al. is ∼9 ms, which would require integration out to a maximum of 27 ms to collect >95% of the charge. Also, an earlier study from the same laboratory (18) reported the use of a time interval of 72 ms for the transient data of rSGLT1.

Hazama et al. (4) also reported that there are no transient currents for voltage steps more hyperpolarizing than −50 mV. We are very confident, however, that this is not the case, as shown in Figs. 2–6.

We also point out that, for a significant period of time now (since at least 1993), articles that discuss the Boltzmann distribution have used the following equation to describe and fit the data (4, 6, 8–11, 13, 15–17, 20): (1)

Lo and Silverman in both of their articles from 1998 (8, 9) used *Eq. 1* except that −*Q*_{hyp} was used in the place of +*Q*_{hyp}, and Zampighi et al. (20) used *Eq. 1* with the *Q*_{hyp} term omitted. The problem with this equation is that as the membrane voltage approaches positive infinity (*V* → +∞), the exponent in the denominator also approaches positive infinity (the fraction approaches zero), and therefore *Q* → *Q*_{hyp}. This is clearly wrong, because *Q*_{hyp} is the charge displaced at hyperpolarizing (negative) potentials. In addition, *Q*_{max} should be −*Q*_{max} because as *V* → −∞, we should have *Q* → *Q*_{hyp} and *Q*_{max} = *Q*_{dep} − *Q*_{hyp}. These are most likely typographical errors that have propagated from a single early source. The fix is simple: replace *Q*_{hyp} with *Q*_{dep} and *Q*_{max} with −*Q*_{max} so that the proper equation is (2)

This change is mostly conceptual and should not affect the validity of any data analysis done with the earlier equation. Note that *Eq. 1* holds when the valence, *z*, is negative. However, *1*) we know of no study of any SGLT1 mutant or species that refers to a negative *z*, and *2*) this case represents a family of curves mirrored about the *x*-axis in relation to the one shown in Fig. 6*A* (a negative *Q*_{dep} and a positive *Q*_{hyp}). The restrictions imposed by investing meaning into the variables *Q*_{max}, *Q*_{hyp}, and *Q*_{dep} with the use of subscripts (and even *V*, by assuming that it is the voltage being jumped to, rather than from) can be avoided by using the more general *Eq. 11* or by using more benign variables such as *A* and *B*. In fact, *Eq. 11* should be used instead of *Eq. 2* because *Eq. 11* contains only three variables (*N*, *z*, and *V*_{0.5}) compared with four (*Q*_{max}, *Q*_{dep}, *z*, and *V*_{0.5}) in *Eq. 2*. This difference is due to the inclusion of the initial potential (or holding potential), *V*_{i}, in *Eq. 11*, which is always known. On the other hand, by fitting for *Q*_{dep} in *Eq. 2*, we are omitting this information (and, in a sense, asking for *Eq. 2* to find it again). In practice we have found only small differences between fits using *Eqs. 2* and *11*.

The average number of expressed carriers per oocyte is 0.34 × 10^{11} (*n* = 18) and is on par with similar calculations for rabbit (6, 20) and human (10, 19) SGLT1 of ∼1 × 10^{11} carriers/oocyte. However, this is an order of magnitude lower than one study for human SGLT1 (hSGLT), which predicted 3.5 × 10^{11} carriers/oocyte (3). These estimates are generally derived at optimal expression levels but can vary depending on the time of observation relative to injection, the viability of individual oocytes, and whether cRNA or cDNA was used.

#### Interpretation of three-state Boltzmann parameters.

In a recent study (7), transient currents were simulated and a set of rate constants and valences for the four-state model shown in Fig. 1 were presented. Table 1 shows *V*_{0.5} and *z* values calculated with these parameters. The valences from the three-state Boltzmann fit and the simulation are almost identical; however, the simulation *V*_{0.5} values are more depolarizing, 55 mV for *V*_{10.5} and 30 mV for *V*_{20.5}. Although we cannot explain this difference in the *V*_{0.5} values, we believe that it is important that the order of the *V*_{0.5} values is intact (i.e., *V*_{10.5} is more hyperpolarizing than *V*_{20.5}). If the three-state Boltzmann fit parameters and the simulation parameters are analogous, then we can say that *z*_{1} and *V*_{10.5} correspond to a conformational change of the unloaded carrier (C_{3} ↔ C_{4} in Fig. 1), whereas *z*_{2} and *V*_{20.5} correspond to an event involving Na^{+} interaction with the transporter (C_{1} ↔ C_{2}).

Meinild et al. (13) found that the Q457C mutant of hSGLT1 labeled with tetramethylrhodamine-6-maleimide produced a fluorescence-voltage curve that could be fit with a two-state Boltzmann equation with *V*_{0.5} = −51 mV and *z* = 0.4. Because these values are virtually identical to *V*_{10.5} and *z*_{1} for wild-type rSGLT1 (Table 1), it is possible that the conformational change monitored by the fluorescent tag at the Q457C site is the same as the electric conformation identified with the three-state Boltzmann fit. In turn, we may be able to associate the quenching of this fluorescent tag with a conformational change of the empty carrier. These correlations, however, are entirely speculative at present and require verification by additional experiments.

#### Turnover rate.

The turnover rate, *k*, represents the number of transport cycles a single SGLT1 protein completes, on average, per second and is typically measured at 100 mM Na^{+}, 10 mM glucose, and −150 mV. An expression for calculating turnover is (3) This expression is derived by dividing the steady-state current, *I*_{ss}, by the number of expressed transporters, *N*, the number of ions moved per transport cycle, *n* (which is almost exclusively set to 2), and the elementary charge, *e*. An equivalent and more generally used form for *Eq. 3* replaces *N* and *e* with *Q*_{max} (maximal charge transfer measured from a Boltzmann fit to the *Q-V* distribution) and *z*_{app} = Σ*z*_{i} (combined apparent valence), also from the Boltzmann fit, (4) so that (5) Using either *Eq. 3* or *Eq. 5* and our results in Table 1, we can calculate the turnover value, *k*, provided that we also know *I*_{ss}. Because we already know that *k* is 22 s^{−1} (9, 15) for a two-state Boltzmann fit, we can estimate the turnover value with the three-state Boltzmann fit parameters to be 22 × (Q_{max}/z_{app})_{2state} × (z_{app}/Q_{max})_{3state} = 22 × N_{2state}/N_{3state} = 34. Therefore, the turnover value of wild-type rSGLT1 is at least 54% larger than previously reported. Of course, based on the estimates of Zampighi et al. (20) that *z*_{app} is ∼3.5 and the hypothesis of Chen et al. (3) and our previous work (7) that a sugar-free model of SGLT1 is composed of at least four states, the turnover rate is likely >34 s^{−1}.

Note that the assumption that *n* = 2 is an ideal based on the Na^{+}:glucose stoichiometry of 2:1. To be more precise, *n* is equal to the average charge moved per transport cycle, and therefore, theoretically, there are a number of factors that will affect a more realistic value for *n*: *1*) it is possible that, on occasion, the transporter will cycle with only one Na^{+}, and therefore *n* would be <2; *2*) other ions may slip through the SGLT1 transport pathway in addition to the two Na^{+} and therefore increase the value of *n* >2; *3*) these non-ideal phenomena may occur in a voltage-dependent manner so that *n* itself could be a function of voltage. Various values have been reported for *n* because there are a number of complicating experimental factors that are not easy to control. On the basis of the reversal potential in the presence of glucose, Chen et al. (2) obtained *n* = 1.96 ± 0.04 (*N* = 18). In another study (12) based on radioactive uptake, it was determined that *n* = 1.9 at −110 mV and *n* = 1.6 at −70 mV. Typically, turnover is measured at a membrane potential of −150 mV, so although *n* may vary, it is probably closer to 2 at this potential.

In summary, our intention with this article has been to bring to light, and to correct, a number of systematic errors in the SGLT1 literature concerning the measurement and fitting of the *Q-V* distribution as follows: *1*) the time domain of the transient currents cannot be discerned by eye, and a technique like the one shown in Fig. 5 should be used; *2*) as many models as possible should be used when fitting *Q-V* data, and the fitting process should take into account quantitative data, such as the residuals or χ^{2}/df when discriminating between the models because visual cues are unreliable; *3*) for step changes in membrane voltage, there are large capacitive currents in the first ∼1 ms that saturate the measuring apparatus, reducing the accuracy of the *Q-V* data, and this can be prevented by replacing the step function with a ramp function of ∼5 ms; *4*) when the signal-to-noise ratio is low, multiple data sets can be added together to increase the accuracy of the data; and *5*) there is a minor error in the two-state Boltzmann equation that has propagated through all previous studies in the literature. Though our focus is on the Na^{+}/d-glucose cotransporter, we seek to discover general principles that can be applied more broadly to the entire class of ion-coupled solute cotransporters. Many transporters and channels have *Q-V* curves that conform to a Boltzmann distribution, and therefore they may benefit from the type of analysis presented here.

Finally, it should be noted that, in regards to transient current and the *Q-V* distribution analysis of SGLT1, there has been an exclusive reliance in the literature on visual cues to determine whether an equation fits the data well. More importantly, there has been little attempt to reconcile differences between the equation used to fit the data and our understanding of the underlying system (i.e., single exponential fit to the transient currents when at least two or three exponentials are expected; *Q-V* data fit with a two-state Boltzmann equation when a three- or four-state fit is expected). We believe this is an important issue that needs to be addressed. Ultimately, achieving a better understanding of the mechanisms involved in SGLT1 function will depend on how the data are explained by the models used to conceptualize transporter activity.

## APPENDIX

### Two- and Three-State Boltzmann Equations

The two-state system in Fig. 7 and the three-state system in Fig. 8 were used in this study to derive equations for a two- and three-state Boltzmann distribution, respectively. The voltage and temperature dependencies of the rate constants shown in Figs. 7 and 8 are given by (6) For later convenience we also used the notation with (7) and where each *z*_{ij} (which has no units) is related to the charge movement across the membrane electric field associated with the pathway from state C_{i} to C_{j} (*z*_{i,i+1} and *z*_{i+1,i} may be different to allow for the possibility of an asymmetric energy barrier); *u* = *F*/(*RT*), where *F* is the Faraday constant, *T* is absolute temperature, and *R* is the gas constant; and *k*_{ij} are rate constants with units of s^{−1}. In addition, the number of transporters in state C_{i} is assigned to the variable *N*_{i}, so the total number of expressed transporters is (8)

It is directly evident from *Eq. 6* that at extreme hyperpolarizing potentials (*V* → −∞), all the transporters move into state one (*N*_{1} → *N*), and conversely, as *V* → +∞, *N*_{2} → *N* or *N*_{3} → *N* for the two- or three-state systems, respectively. At intermediate voltages the system is distributed across multiple states. A Boltzmann curve is an indirect measure of this distribution as a function of membrane potential. The curve is generated by plotting the net charge moved, *Q*, when the membrane potential is changed, over time, from a fixed potential, *V*_{i}, to a variable potential, *V*_{f}.

For the two-state system shown in Fig. 7, the net charge movement for any size voltage change, taking place over a finite period of time starting at *t* = t_{0}, is given by (9) Also, noting that the steady-state occupancy of *N*_{1} is (10) and assuming that the system is at steady state at *t* = t_{0}, *Eq. 9* can be written as (11) Fixing *V*_{i} while leaving *V*_{f} variable produces the standard two-state Boltzmann relation where (12)

In a similar manner we have derived a three-state Boltzmann equation beginning with (13) and (14) to give the following three-state Boltzmann equation: (15)

## GRANTS

This work was supported by a grant to M. Silverman (FRN-15267) as part of the Canadian Institutes of Health Research Group in Membrane Biology (FRN-25026).

## Acknowledgments

We gratefully acknowledge R. Tsushima, P. Backx, and B. Krofchick for helpful discussion.

## Footnotes

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- Copyright © 2004 the American Physiological Society